Counterexamples to maximal regularity for operators in divergence form

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Counterexamples to maximal regularity for operators in divergence form

Journal Article (2024)

Author(s)

Sebastian Bechtel (TU Delft - Analysis)

Connor Mooney (University of California)

Mark Veraar (TU Delft - Analysis)

Counterexamples Divergence form operators Lions’ problem Maximal L-regularity Primary 35K90 Secondary 35B65

DOI related publication

https://doi.org/10.1007/s00013-024-02014-9 Final published version

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https://resolver.tudelft.nl/uuid:a6a126e7-2152-48c9-b696-e5c4a52462a0

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Publication Year

2024

Language

English

Issue number

2

Volume number

123

Pages (from-to)

199-209

Downloads counter

156

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Abstract

In this paper, we present counterexamples to maximal L^p-regularity for a parabolic PDE. The example is a second-order operator in divergence form with space and time-dependent coefficients. It is well-known from Lions’ theory that such operators admit maximal L²-regularity on H^-1 under a coercivity condition on the coefficients, and without any regularity conditions in time and space. We show that in general one cannot expect maximal L^p-regularity on H^-1(R^d) or L²-regularity on L²(R^d).

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